How can I calculate the root mean square speed of carbon dioxide molecules at STP?

Hello!
As we know from the kinetic theory of gases, the mean kinetic energy of gas molecules is proportional to its absolute temperature T.  Because the kinetic energy of a molecule is (m v^2)/2, the root mean square is proportional to sqrt(T). The exact formula is
v_(rms) = sqrt((3RT)/M),
where R approx 8.3 J/(mol * K) is the ideal gas constant and M is the molar mass of a gas. M must be expressed in (kg)/(mol).
 
We know the relative atomic masses of  O and C. They are 16 and 12, and we can compute the molar mass of C O_2: 
12 + 2*16 = 44 (g/(mol)) = 4.4*10^(-2) ((kg)/(mol)).
 
 
STP usually means 0 degrees Celsius, or about 273 K. Now we can obtain the numerical result:
v_(rms) = sqrt((3*8.3*273)/(4.4*10^(-2))) approx 393 (m/s).
 

Single Variable Calculus, Chapter 5, 5.5, Section 5.5, Problem 82

Find the integral $\displaystyle \int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx$

If we let $u = \sin^{-1} x$, then $\displaystyle du = \frac{1}{\sqrt{1 - x^2}} dx$. When $\displaystyle x = \frac{1}{2}, u = \frac{\pi}{6}$. Therefore,



$
\begin{equation}
\begin{aligned}

\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \int^{\frac{1}{2}}_0 \sin^{-1} x \cdot \frac{1}{\sqrt{1 - x^2}} dx
\\
\\
\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \int^{\frac{1}{2}}_0 u du
\\
\\
\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \frac{u^2}{2} |^{\frac{1}{2}}_0
\\
\\
\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \frac{\displaystyle \left( \frac{\pi}{6} \right)^2 }{ 2 } - \frac{(0)^2}{2}
\\
\\
\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \frac{\displaystyle \frac{\pi^2}{36}}{2}
\\
\\
\int^{\frac{1}{2}}_0 \frac{\sin^{-1}x}{\sqrt{1 - x^2}} dx =& \frac{\pi^2}{72}

\end{aligned}
\end{equation}
$

Explain how Jonas's community is hypocritical.

Jonas's community is hypocritical in that it presents itself as a bastion of justice, even as it murders its most defenseless civilians. 
The idea behind "release" is to ensure that physical imperfections and spiritual infractions (which are said to lead to suffering, confusion, conflict, and war) are annihilated. To that end, Jonas' community consistently orders the release of the Old, rule-breakers, and even babies who fail to thrive (especially if they are the weaker of a set of twins). 
In the book, Jonas discovers the true meaning of "release" during his training to become the next Giver or Receiver of Memories. He is devastated when he discovers that his father has been cognizant of the meaning of "release" all along. 
While the Giver is allowed to lie (see Rule no. 8 in Chapter 9), others are supposedly prohibited from doing so. Yet, Jonas's father has obviously lied to Jonas by hiding the truth about "release" from his entire family and others in the community. So, this community is hypocritical because some of its most trusted members are willing to hide their most repugnant acts behind facades of righteousness. Meanwhile, others are more than willing to remain ignorant about the true nature of "release."

What does Gertrudes statement “The lady protests too much, methinks” tell us about her possible guilt in her first husband’s death?

Gertrude's comment "The lady protests too much, methinks" in act 3, scene 2, of Shakespeare's Hamlet exposes her own guilty conscience.
"The lady" that Gertrude is referring to is the player queen. Gertrude makes this comment while watching, along with the rest of the castle, the play that Hamlet has devised for the players to perform, which mirrors the events of his father's murder in hopes that Claudius's response to the uncannily familiar performance will finally confirm Hamlet's suspicions and incriminate his uncle. The player queen is Gertrude's analog in the plot, a caricature of herself, so when Gertrude comments on the other woman's actions her comment reflects back on herself.
The sentiment of "The lady protests too much" is essentially that someone can deny something so many times that it becomes unbelievable. Throughout Hamlet, Gertrude claims ignorance of her new husband/past brother-in-law's vile actions. But here, the audience gets the sense that maybe Gertrude has been protesting too much herself. Maybe she does have her suspicions about what happened to Hamlet's father, and the reason she protests so much is to keep her own doubts at bay.

College Algebra, Chapter 7, 7.4, Section 7.4, Problem 32

Consider the system

$\left\{
\begin{equation}
\begin{aligned}

x + 2y + 6z =& 5
\\
-3x - 6y + 5z =& 8
\\
2x + 6y + 9z =& 7

\end{aligned}
\end{equation}
\right.$

a.) Check that $x = -1, y = 0, z = 1$ is a solution of the system


$
\left\{
\begin{equation}
\begin{aligned}

-1 + 2 (0) + 6(1) =& 5
\\
-3(-1) - 6(0) + 5(1) =& 8
\\
2 (-1) + 6(0) + 9(1) =& 7

\end{aligned}
\end{equation}
\right.
$



$
\left\{
\begin{equation}
\begin{aligned}

5 =& 5
\\
8 =& 8
\\
7 =& 7

\end{aligned}
\end{equation}
\right.
$


It shows that the given value of $x, y$ and $z$ is a solution of the system

b.) Determine the determinant of the coefficient matrix

For this system we have


$
\begin{equation}
\begin{aligned}

|D| =& \left| \begin{array}{ccc}
1 & 2 & 6 \\
-3 & -6 & 5 \\
2 & 6 & 9
\end{array} \right|
\\
\\
|D| =& 1 \left| \begin{array}{cc}
-6 & 5 \\
6 & 9
\end{array} \right| -2 \left| \begin{array}{cc}
-3 & 5 \\
2 & 9
\end{array} \right| + 6 \left| \begin{array}{cc}
-3 & -6 \\
2 & 6
\end{array} \right|
\\
\\
|D| =& 1 (-6 \cdot 9 - 5 \cdot 6) - 2 (-3 \cdot 9 - 5 \cdot 2) + 6 (-3 \cdot 6 - (-6) \cdot 2)
\\
\\
|D| =& -46

\end{aligned}
\end{equation}
$


c.) Without solving the system, determine whether there are any other solutions.

There aren't any other solutions in the given system, since that the lines are unique. Thus, there is only one solution in the system and that is in part (a).

d.) Can Cramer's Rule be used to solve this system? Why or why not?

Cramer's Rule can be used to solve the system because the given system has a determinant of the coefficient matrix.

What is Freud’s theory of Personality Development, and is it still relevant today?

Freud thought that personality developed during one's childhood. This occurred as a child moved through five stages, which he called "psychosexual stages" of development. The stages—oral, anal, phallic, latent, and genital—each involved the ability to satisfy particular needs associated with a certain part of the body.
Because unlimited satisfaction of desires is not conducive to a well-adjusted life, Freud argued that the id, which pursued desire in an animalistic way, had to be controlled by the ego and superego. The extent to which these different parts of one's personality developed determined how well-adjusted a person was. Progression through each stage involved a conflict that had to be resolved in order to develop in a healthy way, and mental illness, such as it was understood, resulted from the failure to resolve these conflicts. For a boy, resolving the strong feelings of attachment to one's mother, for example, was believed to be very important to growth and development. The resolution to the conflict that this entailed was for the boy to emulate his father's masculine behaviors. 
As for the relevance of Freud's theory, most of it has been supplanted by subsequent research and theory. Freud's assumptions about gender, in particular, were very much a product of his own time, and do not represent the conclusions of modern science. But his theory continues to provide a valid framework upon which subsequent research into psychological development has been based.

What are some of the ethical considerations of sending humans into space?

The adventure of space travel readily captures the imagination. Countless books, movies, and television shows have been created around the theme. In practical terms, the scientific and medical advancements spurred by the space program have aided humanity at large. Technologies as diverse as LEDs, advanced artificial limbs, and plane de-icing systems owe something of their creation to the aerospace industry. The possible colonization of Mars will doubtless spark even more innovation. However, such a momentous undertaking cannot ignore the ethical issues raised by sending human beings into space for the time required for the trip. Considerations include physical and mental health as well as the probability of catastrophe and the response to it.
The human skeletal system and microbiome are normally under conditions of consistent gravity and atmospheric protection from solar and cosmic radiation. These will be largely lacking on any prolonged expedition. Muscle weakness and bone demineralization already trouble astronauts. Longer exposure can potentially worsen these issues. Little research has been done regarding the effects of radiation on human gut bacteria, and a healthy microbial colony is essential to proper nutrition and overall health.
With current technologies, any Mars-bound craft will be small and cramped compared with what people are used to on Earth. The psychological balance of the crew is essential. Lack of privacy and close living conditions during the long journey could threaten astronaut mental health and essential group cohesivity.
Finally, the response to an emergency or its practical impossibility cannot be overlooked. It would be extremely difficult, if not impossible, to come to the astronauts' aid in case of emergency, making any dangers faced far starker.
https://spinoff.nasa.gov/Spinoff2008/tech_benefits.html

https://www.space.com/25352-nasa-long-duration-spaceflight-ethical-risks.html